Fractional Nonlinear Partial Differential Equations for Physical Models: Analytical and Numerical MethodsView this Special Issue
New Explicit Solutions to the Fractional-Order Burgers’ Equation
The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables -expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional differential equations (NLFDEs) into ordinary differential equations. In this article, we ascertain the solutions in terms of , , , rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting definite values of the parameters in the general solutions. Mathematical analysis of the solutions confirms the existence of different soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory effect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results.
The nonlinear fractional evolution equations (NLFEEs) emerge frequently in diverse research field of science and applications of engineering. The fractional derivative has been happening in numerous physical problems, for example, recurrence subordinate damping conduct of materials, motion of an enormous meager plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, and controller for the control of the dynamical system. Fractional-order differential equations describe the phenomena. The fractional-order differential equations are broadly used as generalizations of conventional differential equations with the integral order to explain different intricate phenomena in numerous fields including the diffusion of biological populations, electric circuit, fluid flow, chemical kinematics, control theory, signal processing, optical fiber, plasma physics, solid-state physics, and other areas [1–5]. The concepts of dissipation, dispersion, diffusion, convection, and reaction are closely related to the abovestated phenomena, and nonlinear fractional partial differential equations (NLFPDEs) can be used to evaluate them exactly. Wave shape has an effect on sediment transport and beach morpho dynamics, while wave skewness has an impact on radar altimetry signals, and asymmetry has an impact on ship responses to wave impacts. Traveling wave solutions are a special class of analytical solutions for NLFEEs. Solitary waves are transmitted traveling waves with constant speeds and shapes that achieve asymptotically zero at distant locations. The appearance of solitary waves in nature is rather frequent in plasmas, fluids dynamics, solid-state physics, condensed matter physics, chemical kinematics, optical fibers, electrical circuits, bio-genetics, elastic media, etc. Consequently, it is important to search for the exact traveling wave solutions of NLFPDEs to understand the facts. Therefore, many researchers have been motivated on finding the exact solutions to nonlinear fractional-order differential equations, and significant progress has been made in analyzing the exact solutions of these types of equations. The major challenges, however, are that there is no unified numerical or analytical approach that can investigate all sorts of nonlinear fractional-order differential equations. Thus, several numerical and theoretical methods for finding solutions for NLFDEs have been established, for example, the differential transformation method [6, 7], the variational iteration method [8–10], the fractional subequation method , the Kudryashov  method, the homotopy perturbation method [13, 14], the homotopy analysis method , the exp-function method [16, 17], the -expansion method and its various modification [18–22], the Chelyshkov polynomial method [23, 24], the multiple exp-function method , the finite difference method , the finite element method , the first integral method [28, 29], the modified simple equation method , the reproducing kernel method , the two variables -expansion method [32, 33], and the Picard technique .
The time-fractional Burgers’ equation is crucial for modeling shallow water waves, weakly nonlinear acoustic waves propagating unidirectionally in gas-filled tubes, and bubbly liquids. Inc  studied the approximate and exact solutions to the time-fractional Burgers’ equation by the variational iteration method. Bekir and Guner  established the exact solution to the mentioned equation by using the -expansion method. Bulut et al.  examined the analytical approximate solution to the suggested equation through the modified trial equation method. Recently, Saad and Al-Sharif  studied the exact and analytical solutions to this equation. As far as is known, the stated equation has not been investigated through the two variables -expansion technique, exp-function strategy, and expanded tanh function method. Therefore, the aim of this study is to establish further general and some fresh solutions of the abovementioned equation using the suggested methods.
The residual segments of the article is schematized as follows: in Section 2, definition and preliminaries have been introduced; in Section 3, the two variables -expansion method, the exp-function method, and the extended tanh function method have been described. In Section 4, the exact solutions to the suggested equation have established. In Section 5, physical interpretation and explanation of the extracted solutions are provided. In the lattermost part, the conclusions are given.
2. Definition and Preliminaries
Suppose be a function. The -order conformable derivative of is interpreted as for every and . If is -differentiable in some , , and exists; then, . The following theorems point out few axioms that are satisfied conformable derivatives.
Theorem 1. Consider and let us suppose be -differentiable at a point . Therefore,(i), for all (ii), for all (iii), for all constant function (iv)(v)(vi)In addition, if is differentiable, then Some more properties including the chain rule, Gronwall’s inequality, some integration techniques, Laplace transform, Tailor series expansion, and exponential function with respect to the conformable fractional derivative are explained in .
Theorem 2. Let be an -differentiable function in conformable differentiable, and suppose that is also differentiable and defined in the range of . Then,
The Caputo derivative is another important fractional derivative concept developed by Michele Caputo . This definition is particularly useful for finding numerical solutions. The definition of Riesz [40, 41] in relation to the fractional derivative, on the contrary, is also important for extracting numerical solutions. The two concepts are not discussed in depth here since the aim of this article is to establish exact solutions.
3. Outline of the Methods
In this part, we summarize the principal parts of the suggested methods to analyze exact traveling wave solutions to the NLFEEs. Assume the general NLFEE is of the formwhere represents an unknown function, consisting the spatial derivative and temporal derivative , and represents a polynomial of and its derivatives where the highest order of derivatives and nonlinear terms of the highest order are associated. Take into account the wave transformationwhere and are nonzero arbitrary constants.
3.1. The Two Variables -Expansion MethodStep 1: In this subsection, we apply the two variables -expansion method to acquire the wave solutions of the NLFEEs. Take into account the second order ODEs along with the following relations In this manner, it gives The solutions to equation (6) depend on as , and . Case 1: when , the general solution to equation (6) is In view of that, we obtain where . Case 2: if, the solution to (6) is given as follows: Therefore, we obtain where . Case 3: when , the solution of equation (6) is Therefore, we find where and are arbitrary constants. Step 2: in agreement with two variables -expansion scheme, the solution of (5) is presented as a polynomial of and of the form where and are arbitrary constants to be determined later. Step 3: after balancing the maximum order of derivatives and nonlinear terms, which appear in equation (5), it can be fixed the positive integer . Step 4: setting (15) into (5) along with (8) and (10), this modifies to a polynomial in and having the degree of as one or less than one. If we compare the polynomial of similar terms to zero, then it will give a set of mathematical equations which can be unraveled by computational software and finally yield the values of , , , , , and , where ; this condition provides solutions of the hyperbolic function. Step 5: in a similar manner, we can examine the values of , , , , , and , and trigonometric and rational solutions can be established separately for the case of and.
3.2. The Exp-Function Method
Within this section, the key components of the exp-function method are described for searching the traveling wave solution to the NLFDEs. Step 1: the arrangement is to be communicated in the shape as indicated by the exp-function method: where , , and are unknown positive integers, which can be evaluated later, and and are unidentified constants. Step 2: the balancing principle between the highest-order linear and nonlinear terms presented in (5) and substituting (16) into (5) yield and , and the balance of lowest-order linear and nonlinear terms yields the values of and . Step 3: introducing (16) into (5) and setting the coefficient of to zero provides an arrangement of set of mathematical equations for , , , and . Then, unraveling the set with the aid of computer software, such as Maple, we attain the constants. Step 4: substituting the values that showed up in step 3 into (16), we ascertain exact solutions to the NLFEEs in (3).
3.3. The Extended Tanh Function Method
In this section, the suggested extended tanh function method has been interpreted to obtain ample exact solutions to NLFEEs which was summarized by Wazwaz . The basic concept of this method is to present the solution as a polynomial of hyperbolic functions, and then, solving the coefficient of implies solving a system of algebraic equations. The core steps of the extended tanh function method for finding exact analytic solutions of nonlinear PDEs of the fractional order are as follows: Step 1: we consider the wave solution as follows: wherein where is any arbitrary constant. Step 2: taking uniform balance between the maximum order nonlinear term and the derivative of the maximum order appearing in equation (5) to determine the positive constant . Step 3: substitute solution (17) together with (18) into equation (5) with the value of acquired in step 2, which yields the polynomials in . A set of algebraic equations for ’s and ’s are found by setting each the coefficient of the resulted polynomials to zero. With the help of symbolic computational software, namely, Maple, this set of equations for and can be solved. Step 4: inserting the values that appeared in step 3 into equation (17) along with equation (18), we construct closed-form traveling wave solutions of nonlinear evolution equation (3).
4. Analysis of the Solutions
Here, we search further comprehensive exact analytic wave solutions for the stated time-fractional Burgers’ equation by means of the suggested methods. Let us consider the time-fractional Burgers’ equation as follows:where are arbitrary constants. The physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe are described by the time-fractional Burgers’ equation. The fractional derivative results are obtained from the memory effect of the wall friction through the boundary layer. The similar formation can be found in several systems, namely, waves in bubbly liquids and shallow water waves. For equation (19), we recommend the subsequent wave transformation:where be the velocity of the traveling wave. For wave transformation (20), time-fractional Burgers’ equation (19) reduced to the ensuing integral order differential equation:
Integrating equation (21) with zero constant, we obtain
4.1. Solutions through Two Variables -Expansion Method
Considering the homogeneous balance of the highest-order nonlinear term and highest-order derivative showing up in equation (22), the arrangements of equation (15) accept the shapewhere , and are constants to be determined. Case 1: for , embedding solution (23) into (22) along with equations (8) and (10) yields a set of algebraic equations, and by explaining these equations by computer algebra such as Maple, we achieve the following results: Inserting the top values into solution (23), we find the solution to equation (19) in the form where and . Since and are basic constants, one might have picked self-assertively their values. If we take and or in (25), we have where . Case 2: in a comparative way, when , substituting (23) into (22) together with (8) and (12) yields an arrangement of algebraic equations for , , and , and we acquire the following results by working out these equations: The substitution of these results into solution (23) possesses the following expression for the general solution of equation (19): where and . If the unknown parameters are assigned as and and or and in solution (29), it provides the next solitary wave solution: where . Case 3: in the parallel algorithm when , using equations (22) and (21) along with (8) and (14), we achieve a set of mathematical equations whose solutions are
It is substantial to observe that the traveling wave solutions – of the studied equation are inclusive and standard. The attained solutions have not been noted in the earlier study. These solutions are convenient to designate the physical processes of unidirectional propagation of weakly nonlinear acoustic waves via a gas-filled tube, shallow-water waves, and waves in bubbly liquids.
4.2. Solution by the Exp-Function Method
Considering the homogeneous balance, the solution of equation (16) takes the form
Substituting equation (34) into (22) leads a equation in ; here, represents any whole number. Inserting each coefficient of this equation to zero yields a cluster of mathematical equations (for straightforwardness, here, we have discarded) for , , and . These mathematical equations are solved by computer algebra, namely, Maple, which gives the following outcomes:
From the point of view of the above results, we achieve the following generalized solitary wave solutions:
In particular, if and , solution (44) is simplified and offers the kink type solution of the form
The choice of and in (44) gives the singular kink solution:
It is significant to refer that the traveling wave solutions of the considered Burgers’ equation are fresh and standard and were not established in the earlier investigations. It is deduced that physical systems should be assigned of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled tunnel and waves in bubbly fluids.
4.3. Solution Using the Extended Tanh Function Method
The homogeneous symmetry allows solution equation (17) as
Substituting (47) into (22) along with (18) makes the left hand side as a polynomial in . Setting each coefficient of this polynomial to zero, resulting a set of algebraic equations (for simplicity, we have omitted them to exhibition) for , , , , and . Computing the determined set of equations with the assistance of computer algebra, such as Maple, yields the succeeding results:
The solutions established above by the extended tanh approach are advanced and progressive. These might be convenient to describe the relativistic electron and the physical processes of unidirectional propagation of weakly nonlinear acoustic waves via a gas-filled tube.
5. Physical Interpretation and Explanation
In this section, we mainly discuss about the physical interpretation of the determined solitary wave solutions, including kink, singular solitons, singular kink, and periodic wave of the NLFEEs. A graph is an effective approach for explaining mathematical concepts. It is capable of describing any circumstances in a straightforward and understandable manner. This segment explains the incidents by portraying 3D plots of some of the solutions that are found. The portraits are precedents of the solutions shown in Figures 1–6 using the computational software, namely, Mathematica.
The results of the time-fractional Burgers’ equation include the kink soliton, singular soliton, periodic soliton, and some general solitons which are displayed in Figures 1–6. Figure 1 is the kink shape soliton of solution (26) with the values of the parameters , , , , , and within the interval and . The kink soliton is a soliton which rises or descends from one asymptotic state to another as . Solution (51) represents the shape of the plane soliton characterized in Figure 2 for the values of parameters , , , , and within the interval and Solution (31) represents the periodic wave solutions, plotted for , , , , , and within the interval and labeled in Figure 3. When , , , , and , solution (46) represents the singular kink type soliton characterized in Figure 4 within . On the contrary, for the values of , , , , and , solution (53) also represents the kink soliton illustrated in Figure 5 within the interval . Finally, outcome (54) also represents the singular kink soliton for the values of parameters , , , , and within the range , which is labeled as Figure 6. The other figure of the solutions is analogous to the displayed figure; thus, for convenience, these are omitted here.
In this article, using three reliable approaches referring conformable the fractional derivative, we have established scores of advanced, further general, and wide-ranging solitary wave solutions to the time-fractional Burgers’ equation. The ascertained closed-form solutions of the considered equation include kink, single solitons, periodic solitons, singular kink, and some other kinds of solutions, including some free parameters. The obtained solutions are capable to analyze the phenomena of weakly nonlinear acoustic waves propagating unidirectionally in gas-filled tubes, shallow water waves, and bubbly liquids. The dynamics of solitary waves have been graphically depicted in terms of space and time coordinates which reveal the consistency of the techniques used. The accuracy of the results obtained in this study has been verified using the computational software Maple by placing them back into NLFPDEs and found correct. This study shows that all the methods implemented are reliable, effective, functional, and capable of uncovering nonlinear fractional differential equations arising in the field of nonlinear science and engineering. Therefore, we can firmly claim that the implemented methods can be used to compute exact wave solutions of other nonlinear fractional equations associated with real-world problems, and this is our next contrivance.
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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